Asymmetric Statistics of Order Books - The Role of Discreteness and Evidence for Strategic Order...

8/9/2019 Asymmetric Statistics of Order Books - The Role of Discreteness and Evidence for Strategic Order Placement

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arXiv:0906.1387v3

[q-fin.TR]18Ma

y2010

Asymmetric statistics of order books: The role of

discreteness and evidence for strategic order

placement

A. Zaccaria, M. Cristelli, V. Alfi, F. Ciulla, L. Pietronero

May 19, 2010

Abstract

We show that the statistics of spreads in real order books is character-

ized by an intrinsic asymmetry due to discreteness effects for even or odd

values of the spread. An analysis of data from the NYSE order book points

out that traders strategies contribute to this asymmetry. We also inves-

tigate this phenomenon in the framework of a microscopic model and, by

introducing a non-uniform deposition mechanism for limit orders, we are

able to quantitatively reproduce the asymmetry found in the experimental

data. Simulations of our model also show a realistic dynamics with a sort

of intermittent behavior characterized by long periods in which the order

book is compact and liquid interrupted by volatile configurations. The

order placement strategies produce a non-trivial behavior of the spread

relaxation dynamics which is similar to the one observed in real markets.

1 Introduction

The order book is the double auction mechanism [1, 2, 3] which permits toprocess and store the orders placed by investors in a modern financial market.This system is the elementary mechanism of price formation as a consequenceof the arrival of proposals (orders) of buying or selling. There are two classesof orders: market orders and limit orders. The market ones correspond to or-ders to buy/sell at the best available price (called best bid/ask), hence theyare immediately executed. The limit orders instead are orders to buy or sell ata given price which can be not necessarily the best one. By consequence limit

orders may not immediately fulfilled and then they are stored in the order book.The difference between the best ask and the best bid is defined as the spread s.The order prices are not continuous but discrete and expressed in units of ticks.Also the volume of an order is an integer multiple of a certain amount of shares.The mid-price between best ask and best bid can be considered a conventionaldefinition of the price of a stock. The majority of the spread variations are dueto a limit order which falls inside the spread or to a market order which matchesall the orders placed at the best price.In recent years the complex dynamics of the order book has attracted the sci-entific community attention. One of the reasons of this increasing interest isthe availability of a large amount of experimental data which has permitted an

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extensive statistical analysis of the order book properties and has revealed a

number of interesting features and regularities. The main empirical evidencesof the order book are: the fat-tailed distribution of the price of new limit or-ders [4, 5, 6]; the non-trivial power law correlation of the transaction pricesigns [7, 8, 9, 10, 11]; the peculiar shape of the limit order volume distributionwith fat tails and peaked away from the best quote [4, 5]; the non linear responseof the order book to order arrivals [12, 13].Moreover the microscopic market level of the order book shows the stylized factspresent in financial markets at the aggregate level [3, 14, 15, 16]. To interpretthese empirical evidences a series of theoretical models has been introduced[17, 18, 19, 20, 6, 21]. In particular, the models by Farmer et al. [6, 21] show aseven a zero intelligence mechanism can reproduce many of the experimentalfeatures of real order books.In a previous work we have introduced a model [22] where the zero intelligence

paradigm is adopted and in this paper we are going to interpret some experi-mental results in the framework of this model.One of the most challenging issue in this field is the identification of new empir-ical features in order to discriminate and validate the various models proposed.In this paper we investigate the effects induced by the discrete nature of orderbooks. In fact, as we have already noticed, the price of the orders is not a con-tinuous variable but it can only be a multiple of a quantity called tick which is afraction of the currency used in the the market considered. A first consequenceof this aspect is the spontaneous emergence of asymmetries in the system. Forinstance, two configurations of the order book with an even spread or an oddone (in units of ticks) are not a priori equivalent for the mechanism of deposi-tion of limit orders inside the spread. We investigate the fraction of odd spreads

for a data set from the NYSE market finding indeed a strong evidence for thisasymmetry. However, as we are going to see in detail in the paper, the problemis more subtle than expected and also the agent strategies play an importantrole to explain quantitatively this phenomenon. The strategic order placementis also the origin of the non-trivial relaxation pattern observed when a spreadfluctuation takes place.The paper is organized in the following way.In Sec. 2 we show the experimental evidence of the asymmetry of spreads.In Sec. 3 we propose an interpretation of this empirical feature in terms of amicroscopic model we introduced in [22].In Sec. 4 we perform a more detailed data analysis in order to investigate furtheraspect which contributes to the asymmetry.In Sec. 5 our model is properly modified to take into account the new resultsfound in the data analysis.In Sec. 6 a detailed investigation of the role of the agent strategies for the place-ment of order inside the spread is performed in the framework of our model.In Sec. 7 we study the effects of strategic order placements on the spread relax-ation dynamics with respect to a pure zero-intelligence mechanism.Finally conclusions and perspectives are discussed in Sec. 8.

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2 Empirical evidences

In this section we show the empirical evidences that have led us to investigatethe asymmetric nature of the order book.In our analysis of the order book we have considered a data set which spansa period of nearly 80 trading days between October 2004 and February 2005.This data set includes a series of high-frequency (tick-by-tick) information for20 stocks from the NYSE market. These stocks have been chosen to be het-erogeneous in their level of capitalization 1. The information we have for eachstock is the whole list of transactions and quotes, which are the prices of effec-tive deals and of orders respectively. Our data set lacks the information aboutthe whole order book apart of the best bid and the best ask. Therefore a newquote appears only if the best bid or the best ask have been updated. Fromthis data set we have reconstructed the sequence of market and limit orders by

looking to the spread variations. If the spread has increased with respect to itsprevious value we refer to the event as a market order. On the contrary, theevent is defined as a limit order if the spread has decreased. We assume thatthe probability that a cancellation of a limit order may change the spread is sosmall to be neglected. We have also expressed all the prices in units of ticks,so the spread results to be an integer number. At the end of this refinement,our data set is composed only by the series of the events in which the spreadhas changed, each labelled to be a market or a limit order. In this paper werestrict our analysis to limit order events. The complete analysis, including alsomarket orders, will be matter of future researches. To investigate the intrinsicasymmetry that a discrete and finite spread generates, we have firstly analyzedhow the fraction of odd spreads depends on the average value of the spread

for each stock. In Fig. 1 we have plotted, for each stock, the daily fraction ofodd-valued spreads as a function of the average value of the spread. There are80 different points for each stock and also a further average over all the daysis plotted in the inset. We can observe that for almost all points the fractionof odd-valued spreads is larger than 0.5 which would be the expected value ifthe spread was very large (s ). This asymmetry is more marked for stockswith a smaller average spread and it goes diminishing while the average spreadincreases. A small average spread usually corresponds to stocks with a largecapitalization. In the next sections we are going to investigate these results inthe framework of the model we have introduced in a previous work [22].

3 A model for limit order deposition: uniform

case

In this section we propose a simple explanation of the evidences shown in Sec. 2on the basis of the order book model introduced in [22].First of all we briefly recall the main properties of the model. At each timestep an order is placed. This order can be a sell or a buy order with the sameprobability 1/2, and a market or a limit order with probability = 1/3 and1 = 2/3 respectively. A limit order is placed with an uniform distribution

1Market capitalization is defined as the number of shares of a company multiplied by their

price. It is the simplest measure of a companys size.

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2 4 6 8 10 12 14 16 (ticks)

0.4

0.5

0.6

0.7

0.8

0.9

1

fractionofoddspreads

ahavobabrocaidrige

glkgm

jwn

kssmcdmhsmikmlspg

txiudivnowgr

2 4 6 8 10 12 14 (ticks)

0.4

0.5

0.6

0.7

0.8

0.9

1

fractionofoddspreads

average

Figure 1: (Color online) Fraction of odd-valued spreads (in units of ticks) vsdaily average spread for different stocks. We observe a systematic deviationfrom the symmetric case in which the fraction is 0.5. In the inset we plot afurther average over a period of 80 days.

in the interval ]b(t), b(t) + ks(t)] if it is a sell order and [a(t) ks(t), a(t)[ if it isa buy order, where b(t) and a(t) are the best bid and the best ask respectively,s(t) is the spread and k > 1 is a constant. In first approximation k1 is theprobability for a limit order to be put inside the spread (therefore causing aprice change). This dependence on the previous spread value creates a sortof autoregressive mechanism for the order deposition. The tick size and thevolume of the orders are both constant and fixed (q = 1 and = 1). Finally,a cancellation process avoids the divergence of the volume stored in the orderbook.In this framework we can evaluate a number of quantities, for example theprobability that, given a spread s at time t, the new spread s = s at timet + t is even or odd, where t is the time to wait to have a variation of thespread. The dependence on the parameter k is removed because we consideronly the conditioned probability that an event occurs. In this paper we indicatewith s the value of the spread before an incoming event and with s the new valueof the spread consequent to the variation. Here we restrict our analysis only toevents due to limit order arrival. The probability to have an odd spread in the

final state turns to be dependent on the parity of the spread s. Straightforwardcalculations give the probabilities

P(e|o, s) = 12

P(o|o, s) = 12

P(o|e, s) = 12s

s1

P(e|e, s) = 12

s2(s1)

.(1)

where o and e denote if the the spread is odd or even respectively. One cansee that P(o|e) > P(e|e): given an even spread s in the initial state, the nextspread s has a larger probability to be odd rather than even. As expected, bothP(o|e) and P(e|e) tend to 1/2 for s . This simple argument gives a firstexplanation of the empirical evidence of an excess of odd spreads shown in the

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0 2 4 6 8 10 12 14 16 18 20spread (ticks)

0

0.2

0.4

0.6

0.8

1

P(odd|spread)

Figure 2: (Color online) Conditional probability to have an odd-valued spreads given an initial spread s for real data. The plot shows clearly that with highprobability an odd spread is followed by an even one (0.8) and vice-versa.

previous section.

4 Data analysis

Now we intend to give an interpretation of Fig. 1 in view of the results obtained

in the previous section. The dispersion of the points plotted in Fig. 1 can betraced back to the spurious effect introduced by the fact that many differentspreads s give their contribution to the final average. Therefore the next stepis to investigate the frequency of odd spreads conditioned to a given s, in orderto compare Eq. 1 to real data.We can identify the variations of the spread caused by limit orders imposingthe condition s > s because only limit orders inside the spread can decreaseits value (the same argument has been also followed, for example, by [23]). InFig. 2 we show the conditional probability to have an odd spread s startingfrom a spread s, as a function of s. The pattern strongly oscillates around thevalue 1/2: an even spread is most likely followed by an odd one as predicted byEq. 1, but surprisingly also the viceversa is true.

This apparently strange (with respect to the result of Eq. 1) behavior can beattributed to a non-uniform depositions of limit orders inside the spread. Westudied the distribution of the spread variations conditioned to a given valueof the spread s for real data. We found that a consistent fraction 0.7 ofthe limit orders inside the spread is placed at the quote adjacent to the best,as we show in Fig. 3. A reasonable way to model this non-uniform distributionof the limit orders inside the spread is through a piecewise constant function.In this way the probability to put a limit order at the first adjacent quote is, and the probability to put the order in one of the other remaining quotes isequally distributed. This tendency can be also interpreted in terms of agentsstrategies. In fact, the placement an order far from the (previous) best bid or

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0 1 2 3 4 5 6 7 8 9 10 11 12 s (ticks)

0

0.2

0.4

0.6

0.8

P(

s|s)

s = 4s = 6s = 8s = 10s = 12

1 10 s (ticks)

0.01

0.1

1

P(

s|s)

s=4s=6s=8s=10s=12

Figure 3: (Color online) Probability to have a variation s of the spread given aninitial s for different spread values (empirical data). The most probable variationis always s = 1. The probabilities of the other variations (s = 2, 3,...) areweakly decreasing functions and, in first approximation, can be considered asconstant. The dashed line is the power law with exponent 1.8 found in [24] forLondon Stock Exhange. In both cases s = 1 is highly preferred.

ask can be seen as a risky operation in which the agent, by disagreeing withother agents evaluations, tries to trade quickly paying a kind of virtual cost

equal to the distance between the best and her order quote. In other words, anorder near the best is the most conservative position able to change the spread.The next step is to find how much the probability to place a limit order nearthe corresponding best depends on the spread s and on the stock considered. InFig. 4 we plot as a function of s for three stocks which cover a wide range ofcapitalization, finding very different behaviors depending on the stocks and onthe values of the spreads. One possible explaination for this variety of behaviorsis the lack of statistics: in fact, liquid stocks (that is, stocks characterized byfast order execution and small transaction impact on the price) usually havesmall spreads, and hence the statistics for large spreads is poor, and viceversafor illiquid stocks. We can appropriately weight the contribution of differentstocks by averaging on all the data. The resulting curve is approximately aconstant as a function of the spread.

5 A model for limit order deposition:

non-uniform case

In the previous section we have observed a systematic deviation of the experi-mental probabilities P(o|e, s) and P(e|o, s) from the ones of Eq. 1 derived fromthe hypothesis of uniform order deposition. In this section we are going to showthat this discrepancy is mainly due to the non-uniform probability of orderplacement inside the spread, and therefore to agents strategies.In order to include the effect of a non-uniform order deposition, we can gener-

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0 2 4 6 8 10 12 14 16 18 20s (ticks)

0

0.2

0.4

0.6

0.8

1

(s)

brogetxiaverage

Figure 4: (Color online) Probability (s) to place an order at the quote adjacentto the best one as a function of the spread. We have plotted this probabilities forthree representative stocks and also an average over all the 20 stocks. The verydifferent values observed can be explained by considering the different statisticsof the stocks. To properly address this effect we have performed an averageover the 20 stocks of our data set. The result is a value of (s) approximatelyconstant.

alize Eq. 1 in the following way

P(e|o, s) = s1

2

j=1 g(2j|s)

P(o|o, s) = s3

2

j=0 g(2j + 1|s)

P(o|e, s) = s2

2

j=0 g(2j + 1|s)

P(e|e, s) = s2

2

j=1 g(2j|s).

(2)

where g(i|s) is the probability mass function of the deposition mechanism forlimit orders inside the spread and i = 1, . . . , s 1 is the distance from the bestquote of the placement price.In Fig. 4 we have seen that the probability that a limit order produces aspread variation equal to one is weakly dependent on the value of the spread.This suggests a simple approximation for g(i|s) with a piecewise function. Nowwe discuss how to introduce this property in our model.If a limit order falls inside the book or at the best quotes the mechanism for

order deposition is left unchanged. Instead, if a limit order is placed insidethe spread, the probabilities associated to the s 1 available quotes are nomore uniform but highly peaked around the quote adjacent to the best. Thedeposition probabilities for a buy or a sell order, for a given s, become

g(i|s) =

g(1|s) =

g(i|s) = 1s2

i = 2,...,s 1.(3)

For buy orders the index i is the distance from the best bid while for sell ordersthe index i is the distance from the best ask. In the previous section we haveseen that in real markets can be approximately considered as constant and

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its value is about 0.7. Clearly Eq. 3 is meaningful only for s 3 since for s = 1

limit orders cannot fall inside the spread and for s = 2 a limit order inside thespread is always placed at the quote adjacent to the best one.It follows directly from Eq. 3 that the transition probabilities now read

P(e|o, s) = + 12s3s2

P(o|o, s) = 12

s1s2

P(o|e, s) = + 12

P(e|e, s) = 12

.(4)

In Fig. 5 we plot a comparison between the expression of Eq. 4 and thecorresponding probabilities evaluated from our data set. The experimental re-sults are obtained by averaging over all the 80 trading days and over all the 20stocks. The oscillating behavior can be explained by considering that a vari-

ation of spread of one tick (s = 1) is highly preferred with respect to othervariations. Hence an odd spread goes more likely to an even one and vice-versa.We are now able to separate the two effects that contribute to enhance the frac-

2 4 6 8 10 12 14 16 18 20spread (ticks)

0

0.2

0.4

0.6

0.8

1

P(

odd|spread)

modelexperimental

Figure 5: (Color online) Comparison between the experimental data and thephenomenological model described in the text. The phenomenological proba-bilities of the model (dots) show a small systematic overestimation of the os-cillations with respect to the experimental ones (triangles). This effect can beeasily understood in terms of the approximation considered in Eq. 3.

tion of odd spreads and produce the pattern of Fig. 1 through a simple MonteCarlo simulation. These two contributions are the intrinsic asymmetry due todiscreteness and non-uniformity of order deposition. As initial conditions wegenerate some sequences of spreads with different means, in order to representdifferent virtual stocks. Starting from each sequence we simulate the transitionto odd or even spreads s according to the probabilities of Eq. 1 and Eq. 4. Insuch a way we can evaluate the average fraction of odd spreads for each virtualstock. In Fig. 6 we compare the empirical average asymmetry with the resultsof the Monte Carlo simulations in the two cases just mentioned. The intrinsicasymmetry alone is not able to fit properly the experimental data which instead

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are well reproduced by considering the two combined effects. Nevertheless we

can observe some deviations for large spreads (> 6). This is due to the fact thatwe have assumed a constant probability to place order at quotes different fromthe one adjacent to the best. When the spread grows the error introduced bythis assumption becomes larger. It is worth noticing that here we are neglecting

2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

fraction

ofoddspreads

average (NYSE data)uniform - Eq. (1)

=0.7 - Eq. (4)

Figure 6: (Color online) Fraction of odd-valued spread vs average spread forexperimental data and for two different Monte Carlo simulations. The first sim-ulation (squares) is performed using the uniform order deposition mechanism.The second one (crosses) is instead performed with the non uniform mechanism.These two simulations permit to investigate the two contributions to the asym-

metry between odd and even spreads. As expected we observe that the intrinsicasymmetry does not reproduce the experimental pattern (dots) while the twocombined effects fit the experimental data very well. The small discrepancyfound for large spreads is originated by the approximation made in Eq. 3.

the correlations between the spread values. Anyway we can recover the experi-mental behavior even with this uncorrelated sequence since we are averaging ontimes far longer than the time scales of the spread correlation.

6 The effects of the strategic deposition of or-

dersAn interesting question concerns the role of the parameter and how the non-uniform deposition inside the spread affects the order book statistical properties.Our model allows this kind of investigation and permits to study the effect ofdifferent strategies of order placement inside the spread.A numerical simulation reveals that if the same set of parameters of Sec. 3 andof [22] is used, the spread dynamics diverges for > 0.85. To explain this effectwe have to consider the interplay between market and limit orders. Marketorders tend to move away the two best quotes eroding the book. Instead limitorders tend to reduce the spread by coupling the processes followed by best askand best bid. In such a way the process for the spread is somehow stationary.

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In this framework the deposition rules play an important role in softening or

strengthening the coupling action of limit orders. In fact the coupling betweena(t) and b(t) is ruled by two elements. The first one is the fraction of orderswhich fall inside the spread. A larger fraction of these orders produces a strongercoupling between the best quotes. The second is the mechanism of deposition oforders inside the spread, which governs the average spread variation producedby a limit order. In order to analyze the effect of the deposition mechanism wecan reason as it follows. By considering the symmetry of the uniform case, wefind

< s >

s=

1

2(5)

while in the non-uniform case, from Eq. 4, we obtain

< s >

s=

s+

(1 )[s(s 1) 2]

2s(s 2). (6)

We recall that s 3 since for s = 1 limit orders cannot be placed inside thespread and for s = 2 we always have s/s = 1/2. The inequality < s > /s 1/2 for Eq. 6 is satisfied when

2 s + 1

(7)

and we observe that ( + 1)/ > 3 only for < 0.5 (see Fig. 7). Consequentlythe average spread variation and the coupling action of limit orders in the non-uniform case are never larger than the one produced by the uniform mechanismin the region of parameters investigated (s 3 and > 0.5).

Fixed 0.8 we can analyze the properties of our model and we will come backon the problem of the stability with respect to at the end of this section. Theset of figures in Fig. 8 clarifies the role of a non-uniform deposition inside thespread. In Fig. 8a and Fig. 8b we have plotted the probability density functionsfor the price variations (returns) and for the spreads respectively. We observethat in the non-uniform case the system produces larger fluctuations and largeraverage spreads, as we expected from the previous discussion. It is interestingto notice that when the system tends to a regime in which the order book isalways compact, i.e. in which most of the quotes inside the book are occupied,the statistical properties becomes nearly independent on the deposition details( 0.25).Fig. 8c reveals that the non-uniform deposition also amplifies the fluctuationsof the granularity g, defined as the linear density of the volume stored in a

side of the order book [22]. In particular the non-uniform deposition shows anon-trivial temporal structure for granularity that resembles an intermittencyphenomenon. Since most of the arriving limit orders are placed adjacent tothe best quote the order book stays for long times in a quiet and compactstate characterized by an average spread whose value is nearly one. This isthe dominant regime of our simulated order book, but sometimes bursts ofvolatility are observed. In fact, when a large fluctuation of spread occurs, theautoregressive property and the non-uniformity of the limit order depositionmake the relaxation towards the compact state very slow. This intermittencyis directly related to the volatility correlation that is far longer in the non-uniform case than in the uniform one (see Fig. 8d). We want to stress that a

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

2

3

4

5

6

7

8

spread

s=2s=(+1)/

/s < 0.5

/s>0.5

Figure 7: Phase diagram of the relative average spread variation for the non-uniform mechanism. The relative average spread variation < s > /s is 0.5for the uniform case. < s > /s is larger than 0.5 only for spreads includedin the region between s = 2 and s = ( + 1)/. The highlighted region (s 3and > 0.5) corresponds to a realistic scenario. We see that in this regionthe relative average variation of the spread produced by a limit order is alwayssmaller than 0.5. Therefore the non-uniform deposition reduces the couplingaction of limit orders with respect to the uniform case.

small volatility clustering is already present in the uniform case. Its origin canbe traced back to the dependence on the past spread values of the depositionmechanism. This simple effect introduces an exponential correlation and sofixes a characteristic time-scale. The non-uniform deposition instead amplifiesthis effect because the mechanism sets a further and longer time-scale thatdepends on (the correlation length increases for increasing values of ). Thecorrelation functions in Fig. 8d obey to an exponential decay except for shorttime lags where some spurious effects take place. The bursts of volatility ofthe non-uniform case are even more evident if we represent the complete orderbook. In Fig. 9, that corresponds to the uniform case, the order book is alwayscompact. Instead in Fig. 10 we plot the non-uniform case and we find that thesystem stays for most of the time in a regime which is very similar to the oneof Fig. 9 (regions I,III,V) but sometimes regions characterized by large spreads

and large price movements appear (regions II,IV,VI). It is worth noticing thatin this model large price fluctuations emerge spontaneously, being triggered bya random spread variation (and vice-versa). This mechanism resembles thephenomenon of self-organized criticality [25, 26].It can be argued that also for larger values of the uniform case could producethis intermittency because this correspond to an increase of the time-scale onwhich the autoregressive mechanism is able to produce local volatility clusteringas we can observe in Fig. 8d. Nevertheless the correlation is still far shorterwhen = 0.33 with respect to the one generated by the non-uniform case with = 0.3. Furthermore the magnitude of the correlation is smaller than the oneof the non-uniform case with lower values of. Finally a visual inspection of the

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1 10 100returns

10-5

10-4

10-3

10-2

10-1

100

pdf(returns)

=0.7 =0.33

=0.7 =0.30

=0.7 =0.25

unif =0.33

unif =0.30

unif =0.25

a)

0 5 10 15spread

0

0.1

0.2

0.3

0.4

0.5

pdf(spread)

=2.77 =0.7 =0.33

=2.17 =0.7 =0.30

=1.71 =0.7 =0.25

=15.8 unif =0.33

=2.07 unif =0.30

=1.80 unif =0.25

b)

0 5 10 15 20 25g

0

0.1

0.2

0.3

pdf(g)

=0.7 =0.33

=0.7 =0.30

=0.7 =0.25

unif =0.33

unif =0.30

unif =0.25

c)

0 200 400 600 800 1000time (ticks)

0

0.05

0.1

0.15

0.2

Autocorrelation|r|

=0.7 =0.33

=0.7 =0.30

unif =0.33

unif =0.30

d)

Figure 8: (Color online) Statistical properties of the simulated order book forthe uniform and non-uniform case. In panels a) and b) we plot the probabilitydensity functions for the returns (p) and spreads respectively for differentmarket order rates (). These plots show that the average fluctuations of spreadsand returns are larger in the non-uniform case. When the order book turns toa compact regime ( 0.25), the statistical properties of the model becomenearly independent on the deposition details. Panel c) reveals that the non-uniform deposition produces non-trivial fluctuations of liquidity/granularity. Inpanel d) we plot the autocorrelation of the absolute values of returns and weobserve the presence of persistent volatility. The decay of the autocorrelation

function of the absolute values of returns is exponential except for short timelags where spurious effects take place. This persistent behavior suggests thepresence of an intermittent dynamics for the order book characterized by burstsof volatility.

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order book reveals that an intermittency of a kind appears but it is very small

and we never obtain an order book with sudden transitions from a compactregime to a volatile regime as it happens in region II of Fig. 10.

Figure 9: (Color online) Snapshot of the simulated order book in the uniformcase ( = 0.3). The order book is always in a compact regime in which the aver-age spread is nearly 1 and a small and local volatility clustering is observed dueto the autoregressive deposition rules. For higher values of small deviationsfrom the compact regime appear but these phenomena cannot be compared tothe intermittency produced when the order deposition is non-uniform.

Now we discuss the stability of the model for 0.8. We have seen in Figs.8 a,b that, in order to make the system stable with respect to , a possiblesolution is the reduction of the probability of market orders . In such a way itis possible to increase the average length of limit order sequences and then tocompensate the lower coupling. However increasing values of (> 0.8) wouldimply a choice for 0.25 (or even smaller) and in this region of parametersthe order book is always compact and all volatility bursts disappear. Howeverit is worth noticing that these ranges of parameters are usually not observed inempirical data.

7 Spread relaxation: role of the strategic orderplacement.

Ponzi et al. in [24] studied the relaxation dynamics after an opening or a closingof the spread in LSE 2 order book. They find a slow relaxation of the spreadtowards the mean value. This decay is compatible with a power law and theauthors argue that the absence of a characteristic time scale is due to the pres-ence of a strategic placement of the orders.

2London Stock Exchange

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Figure 10: (Color online) Snapshot of the simulated order book in the non-uniform case ( = 0.3 and = 0.7). These deposition rules produce an orderbook which is typically quiet and compact as in the uniform case (region I, III,V) but exhibits bursts of activity due to large fluctuations of the spread (regionII, IV, VI). The system gives rise to a sort of intermittency since volatility isvery persistent and clustered.

We want to verify this empirical findings in the framework of the model intro-duced in the previous sections. In this respect we define, as in [24], the quantity

G(|) = E[s(t + )|s(t) s(t 1) = ] E[s(t)] (8)

where E[] is the average on the whole time serie, is the spread variationoccured at time t (i.e. = 0) and 0. It is worth noticing that, in ourmodel, is expressed in time event units differently from the analysis in [24]which is performed in physical time. The mapping between these two time unitsis not necessarily linear and therefore a quantitative agreement should not bealways expected.We perform the spread relaxation analysis in both the uniform case and non-uniform case and we report the results in Fig. 11. The non-uniform mechanismproduces a plateau of a kind and then for > 100 a faster decay to normalspread values. Instead in the uniform case the spread relaxation is much faster

than the previous case. Now we analyze in detail the non-uniform case. InFig. 12 we report the function G(|) for positive and negative values of (corresponding respectively to openings and closings of the spread at time =0). As in [24], we observe two slightly different patterns for negative and positive. Instead, we do not observe such a difference in the uniform case, this meansthe nature of the relaxation dynamics is completely different for these two cases.We can conclude that, in order to obtain a realistic spread relaxation function,a simple change to a pure zero-intelligence model consists in the non-uniformmechanism of limit orders deposition inside the spread described in section 5.

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100

101

102

103

time (event time)

10-2

10-1

100

G(|)(ticks)

=4 unif=2 unif=4 =0.7=2 =0.7

Figure 11: (Color online) Spread decay given in the uniform case (solid lines)and in the non-uniform case (dashed lines). We observe a much slower decay inthe latter case.

8 Conclusions and perspectives

The order book is a system which is intrinsically discrete, for instance the quotesof placement of an order must be a multiple of the tick size. We have investi-gated which are the effects of this discreteness finding non-trivial aspects anddeviations with respect to a continuous regime.

The starting point of this work has been the observation that odd and evenspreads are not equivalent for limit order deposition when the available quotesinside the spread are discrete. In fact, if a uniform deposition of orders insidethe spread is assumed, the system spontaneously prefers odd spreads.One of results of this paper confirms that this asymmetry is present in real orderbooks and that the fraction of odd spreads is significantly above 0 .5. Howeverthe asymmetry observed cannot be explained quantitatively only by consider-ing the discrete nature of the order book. Indeed we have found that a secondeffect also contributes to modulate the asymmetry, the fact that agents preferthe quote adjacent to the best one when they place orders inside the spread.Both these contributions have been investigated in the framework of a micro-scopic model introduced in a previous work [22]. The model permits to compare

the effects of uniform and non-uniform deposition mechanisms for limit ordersinside the spread. We have found that the asymmetry can be quantitativelyreproduced in the framework of our model by introducing a non-uniform depo-sition mechanism.Another result is the emergence of a sort of intermittent dynamics in which aregime characterized by a compact and liquid order book dominates but burstsof volatility also appear. This intermittent behavior is also observed in realorder books. In this respect, in the framework of our model, we compare theuniform and the non-uniform mechanism for order deposition with respect tothe dynamics of the spread relaxation when a fluctuation occurs. We find thatthe introduction of a simple rule of order placement is sufficient to reproduce

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100

101

102

103

time (event time)

10-1

10

0

101

G(|)(ticks)

=10=9=8=7=6=5=4=3=2

100

101

102

103

time (event time)

10-2

10-1

100

101

G(|)(ticks)

=10=9=8=7=6=5=4=3

Figure 12: (Color online) Spread decay in the non-uniform case for positive

(top panel) and negative (bottom panel) values of . The value of G(|) ishigher for higher absolute values of . The patterns found in our model arevery similar to the empirical results of Ponzi et al. in [24].

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the peculiar pattern observed in real data (see [24]). The observed relaxation

cannot be explained by a pure zero-intelligence model (see also [27]).An interesting point is how the agents strategies take into account the discreteproperties of the order book. These question will be matter of future works.

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